Part 2: Entropy and Phase Transformations 34
4.2 Vibrations in Nanocrystals
Vibrations of small crystals have been studied for many years. Calculations for both isolated nanocrystals [27, 28] and nanocrystals with rigid constraints [29] established the characteristic en- hancement in the number of modes at low energies that has been observed in all subsequent exper- iments and calculations of the DOS for nanomaterials. This trend implies that the extra spectral weight at low energies originates from degrees of freedom in the microstructure, which may include vibrations that tend to localize at grain boundary regions or propagate along them, motion of the nanocrystals with respect to each other, or some combinations of these degrees of freedom. Investi- gation into the exact nature of these additional modes at low energies, including their dimensionality and origin, has been the subject of frequent study over the last 15 years.
The establishment of the NRIXS technique has lent considerable experimental weight to the quantitative understanding of these spectral enhancements owing to its inherent low background[30, 31, 32,33, 34, 35,36, 37, 38, 39, 40]. Refinement of theoretical tools, namely molecular dynamics
g(E) (eV-1 )
3 4 5 6 7 8 9
10 Energy (meV)
200
150
100
50
0 g(E) (eV-1 )
50 45 40 35 30 25 20 15 10 5
0
Energy (meV)
Figure 4.1: Bulk bcc Fe (solid line) and Cu-5.6 at.% 57Fe with ∼28 nm iron crystallites. The characteristic enhancement of the phonon DOS at low energies and broadening of spectral features is evident from comparison of these two spectra.
simulations and techniques to define symmetry within a particle, has also led to the continual development of this field.
The earliest experimental reports of changes to the phonon DOS for nanocrystalline materials were from three independent inelastic neutron scattering measurements performed in 1995. Suzuki and Sumiyama reported hints that the high-frequency components of phonon spectra are altered in nanocrystalline Ta [41], but described no general trends. The large enhancement of the phonon DOS at low energies was reported for Ni3Al with a 7 nm crystallite size, although it was suggested at the time of the report that this enhancement could have been caused by quasielastic scattering by hydrogen in the material. A study of nanocrystalline Ni also showed a strong enhancement in the DOS below 15 meV for compacted nanoparticles when compared to polycrystalline Ni. [42]
A subsequent inelastic neutron scattering measurement the following year corroborated the Ni3Al result [43]. Measurement of 12 nm and 28 nm nanocrystalline Fe, which is not expected to absorb a significant amount of hydrogen, definitively established the enhanced DOS at energies below 15
features, is shown in Fig. 4.1for bulk bcc iron in comparison with results for the Cu-5.6 at.%57Fe presented in Section4.3. The nanomaterial here is for Fe crystallites∼28 nm in size embedded in a Cu matrix.
The distortions of the phonon DOS are larger for materials with smaller crystals. In particular, the enhancement of of phonon DOS at low energies increases with the inverse of the crystallite size.
This size-dependency was demonstrated for inelastic neutron scattering measurements of nanocrys- talline fcc Ni3Fe, with sizes ranging from 6 to 50 nm. [32] Nanocrystalline nickel-iron powders were synthesized by mechanical alloying and subjected to heat treatments to alter their crystallite sizes and internal strains. It was determined that the enhancement of the phonon DOS at low energies decreased with crystallite sizedasd−1. The coefficientα, obtained from a fit of the DOS below 15 meV to the Debye law
g(ε) =αε2 (4.3)
plotted against inverse grain size shows a linear relationship. This relationship suggests that the phenomena at low energies is related to the surfaces of the crystallites, such as surface modes;
however, the authors caution against over-extrapolating these results, as the frequency spectrum of surface modes should depend on the nature of the contacts between crystallites, and these contacts probably change during the annealing process used to alter particle size.
This same size effect was also reported for bcc iron nanoparticles prepared by inert gas conden- sation and measured in nuclear resonant inelastic x-ray scattering [34], This work speculates that oxidation, in addition to interface vibrational modes, plays a role in the enhancement of low-energy modes, but affirms the increase in low-energy modes with decreasing particle size. A plot of the coef- ficientα, calculated from the fit of the DOS to Eq.4.3in the 1.5 -10 meV range, against a calculated
fraction of iron particles that are pure bcc (obtained by subtracting the fraction of oxidized atoms from one) shows a strong correlation between the coefficient αand fraction of pure iron particles, though it is not quantitatively linear. The authors note that the decrease in particle size tends to raiseαsolely by increasing the fraction of interfacial sites, speculating that the DOS is thus a sum of a partial DOS of interfacial atoms that depends on particle size and a partial DOS of crystalline atoms that is independent of particle size.
For consolidated nanocrystals, the enhancement in the phonon DOS at low energies does not extend to arbitrarily low energies and arbitrarily long wavelengths. Measurements of phonons in the micro eV energy range performed found the phonon DOS of nanocrystalline bcc Fe and fcc Ni3Fe to be enhanced at micro eV energies compared to bulk samples, but the enhancement was markedly greater at meV energies [31]. The enhancement in the DOS at micro eV energies is more charac- teristic of long waves in a homogeneous medium resulting from a reduction in sound velocity. This suggests that the enhancement in the meV energy range originates with features of the nanostruc- ture, while the long-wavelengths of the micro eV energies are involved in the cooperative motions of many nanocrystals. The compact microstructure of nanocrystals forms a coupled dynamical sys- tem, but surface modes are still expected when the grain boundaries have altered densities of force constants. The change in enhancement between the two energy ranges is the characteristic behav- ior of a ‘confinement effect’ where there are no phonon modes below a long-wavelength low-energy cutoff. The cutoff of surface modes at long wavelengths is qualitatively consistent with the smaller enhancement of the DOS at micro eV energies.
Many experiments have sought to gain further insight into whether these low-energy excitations of nanocrystalline materials show characteristics of reduced dimensionality. In the continuum limit, all sound waves have linear dispersions withε/¯h=ω∝k, so in three dimensions,g(ε)∝ε2. Fitting to g(ε) to obtain the exponentnofg(ε)∝εn is the common method for establishing dimensionality.
In this context, the physical nature of the low-E excess modes in nanocrystalline materials has been reported to be linear [44], nonlinear [45, 46,47], and Debye-like quadratic [48, 42,43,30, 32, 33,34, 31,49]. Theoretical work investigating the origin of low-energy modes in nanoparticles has
from three types of atoms determined by topological short-range analysis.[46] These calculations determined low-energy enhancement to be caused mainly by the grain boundaries, with internal surfaces playing only a minor role, and atoms inside the grains retaining the DOS of bulk material.
Experimental investigation attempting to distinguish the vibrational properties of the interiors of nanocrystalline materials and their interfaces with other particles were performed for NRIXS measurements of Fe90Zr7B3 with particle sizes between 2 and 15 nm.[49] The relative fractions of interfaces and interiors were determined by conversion electron Mossbauer spectroscopy in which the spectra were decomposed into contributions from atoms located in nanoparticle interiors, at the surfaces of the interior, and at grain boundaries. This work assumed that the DOS could be decomposed into contributions from each of these regions, the relative weights of which were determined from the Mossbauer spectra. This experiment corroborated the previous theoretical work, finding that the anomalous enhancement of the DOS at low energies originates from the DOS of the interfaces and grain boundaries, with the DOS from the nanoparticle interiors resembling that of the bulk. Further, it was determined that the enhancement of the DOS at low and high energies scales linearly to the atomic fraction of interface atoms.
The origin of the low-energy mode enhancement in nanoparticles has remained an open question for which numerous experiments have been undertaken to identify the specific origins of the unique phonon spectra of nanocrystals, and to decouple competing effects including the nature of surface atoms, low-coordinated interfacial atoms, and oxide surfaces [39,35,40]. The most recent experimen- tal work to address this question was performed on supported, isolated, size-selected nanoparticles that were capped with a Ti layer to prevent oxidation, representing the most thorough attempt to decouple competing effects including the nature of surface atoms, low-coordinated interfacial atoms, and oxide surfaces[50]. The surprising result was found that 3D-Debye behavior was observed for nanoparticles with an average height of less than 2 nm, but non-Debye behavior,n=1.4, was found
for slightly larger nanoparticles ranging from 2.6-6 nm in height. The authors conclude that this result can be explained by considering the larger nanoparticles to be polycrystalline, thus contain- ing grain boundaries in their interior, while the smaller nanoparticles have a single grain structure.
Thus, the reduced effective dimensionality in the larger particles is due to low-energy vibrational modes at grain boundaries.